Vol. 35, No. 1, 1970

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On a six dimensional projective representation of the Hall-Janko group

John Hathway Lindsey, II

Vol. 35 (1970), No. 1, 175–186

It is shown that there is a unique group G with property I : G is a central extension of Z2 by the Hall-Janko group of order 604,800 in which a 7-Sylow subgroup S7 is normalized by an element of order four. Also, G has a six-dimensional complex representation X. The proof is rather round-about. First, it is shown that there are at most two six-dimensional linear groups X(G) projectively representing the Hall-Janko group, and all such linear groups are algebraically conjugate. The character table and generators found by M. Hall for G∕Z(G) are used. It is shown that a linear group L over GF(9) coming from the one candidate for a six dimensional group projectively representing the Hall-Janko group actually satisfies property I. This is done by showing that L has a permutation representation on 100 three-dimensional subspaces of (GF(9))6 and the image is permutation isomorphic to Hall’s permutation group. Hall later studied the geometry of these subspaces. In the course of constructing the character table of any group G with property I,G was found to have a sixdimensional representation. Once this representation is known to exist, it is possible to give two easier ways of constructing generators. The faithful characters on G are given in the appendix with only one representative of each pair of algebraically conjugate characters.

Mathematical Subject Classification
Primary: 20.29
Received: 4 February 1970
Published: 1 October 1970
John Hathway Lindsey, II